Properties of graphs with large girth

Abstract:

This thesis is devoted to the analysis of a class of
iterative probabilistic algorithms in regular graphs, called
locally greedy algorithms, which will provide bounds for
graph functions in regular graphs with large girth. This class is
useful because, by conveniently setting the parameters associated
with it, we may derive algorithms for some well-known graph
problems, such as algorithms to find a large independent set, a
large induced forest, or even a small dominating set in an input
graph G. The name ``locally greedy" comes from the fact that, in
an algorithm of this class, the probability associated with the
random selection of a vertex v is determined by the current
state of the vertices within some fixed distance of v.
Given r > 2 and an r-regular graph G, we determine the
expected performance of a locally greedy algorithm in G,
depending on the girth g of the input and on the degree r of
its vertices. When the girth of the graph is sufficiently large,
this analysis leads to new lower bounds on the independence number
of G and on the maximum number of vertices in an induced forest
in G, which, in both cases, improve the bounds previously known.
It also implies bounds on the same functions in graphs with large
girth and maximum degree r and in random regular graphs. As a
matter of fact, the asymptotic lower bounds on the cardinality of
a maximum induced forest in a random regular graph improve earlier
bounds, while, for independent sets, our bounds coincide with
asymptotic lower bounds first obtained by Wormald. Our result
provides an alternative proof of these bounds which avoids sharp
concentration arguments.
The main contribution of this work lies in the method presented
rather than in these particular new bounds. This method allows us,
in some sense, to directly analyse prioritised algorithms in
regular graphs, so that the class of locally greedy algorithms, or
slight modifications thereof, may be applied to a wider range of
problems in regular graphs with large girth.